https://file.aiquickdraw.com/mathbot/file/172921414154199dbikzg.jpg

Let's go through the problem in number 9 step by step. Here's what it asks:

### Given table:
| Quantity Sold and Produced | Costs   | Revenue | Profit |
|----------------------------|---------|---------|--------|
| 600                        | $9,350  | $25,000 | $15,650 |
| 1200                       | $18,350 | $45,600 | $27,250 |
| 1800                       | $27,350 | $57,650 | $30,300 |
| 3000                       | $45,360 | $60,000 | $14,640 |

Now, let’s solve the subquestions one by one:

### 9a) **Fill in the table and identify at which level profit is maximized.**

We are asked to identify the quantity of items where profit is maximized based on the table.

- **Profit** = Revenue - Costs
- The profits are already calculated and filled in for each quantity level:
  - For 600 units: Profit = $25,000 - $9,350 = $15,650
  - For 1200 units: Profit = $45,600 - $18,350 = $27,250
  - For 1800 units: Profit = $57,650 - $27,350 = $30,300
  - For 3000 units: Profit = $60,000 - $45,360 = $14,640

**Profit is maximized at 1800 units sold**, where the profit is $30,300, as correctly indicated.

### 9b) **What is the average rate of change in Profit between 1200 and 1800 units sold?**

The formula for the average rate of change is:
\[
\text{Average rate of change} = \frac{\Delta \text{Profit}}{\Delta \text{Quantity}}
\]
Here:
\[
\Delta \text{Profit} = 30,300 - 27,250 = 3,050
\]
\[
\Delta \text{Quantity} = 1800 - 1200 = 600
\]
So, the average rate of change in profit is:
\[
\frac{3050}{600} = 5.08
\]
This means the profit increases by $5.08 for each additional unit sold between 1200 and 1800 units.

### 9c) **Find a quadratic regression equation to model profit and the associated marginal profit function.**

You are asked to model the profit as a quadratic function $ P(x) $ based on the given data points. The profit values at $ x = 600, 1200, 1800, 3000 $ can be used to estimate this quadratic function of the form:
\[
P(x) = ax^2 + bx + c
\]
Where:
- $ P(600) = 15650 $
- $ P(1200) = 27250 $
- $ P(1800) = 30300 $
- $ P(3000) = 14640 $

From your work and based on the hint about Desmos, you already found the regression equation:
\[
P(x) = 3050 - 1200x^2 + 1800x
\]
This quadratic function models how the profit changes with the number of units sold.

The **marginal profit function** is the derivative of $ P(x) $. If $ P(x) = ax^2 + bx + c $, then:
\[
P'(x) = 2ax + b
\]
Substituting the values for $ a $ and $ b $ from the regression equation, you would get the marginal profit function.

### 9d) **Profit is maximized when $ x = 4 $.**

It seems you calculated that the maximum profit occurs at $ x = 4 $. Without further clarification, this likely refers to a specific point on a scaled graph that was obtained through further graphing or algebraic work.

Would you like further clarification on this part, or should I help derive the derivative for marginal profit directly?

---

Let me know if you would like any additional details or clarifications. Here are 5 related questions to consider next:

1. How is the quadratic regression formula derived step-by-step for a dataset like this?
2. What does the marginal profit tell us about how profit changes with each additional unit sold?
3. Can you explain how second-order derivatives can help in determining whether profit is maximized?
4. How can profit models like this be used in real business decision-making processes?
5. What other forms of regression (e.g., linear) could we use to model different types of business data?

**Tip:** Understanding the relationship between cost, revenue, and profit can help businesses maximize profitability by finding the optimal balance between production and pricing strategies.

Learn how to solve a business problem involving quadratic regression and profit maximization. This example covers the calculation of average rate of change, deriving the quadratic equation for profit, and determining when profit is maximized based on units sold.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation